1 / 30

Section 3.2

Math in Our World. Section 3.2. Truth Tables. Learning Objectives. Construct truth tables for negation, disjunction, and conjunction. Construct truth tables for the conditional and biconditional. Construct truth tables for compound statements. Identify the hierarchy of logical connectives.

cutler
Télécharger la présentation

Section 3.2

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Math in Our World Section 3.2 Truth Tables

  2. Learning Objectives • Construct truth tables for negation, disjunction, and conjunction. • Construct truth tables for the conditional and biconditional. • Construct truth tables for compound statements. • Identify the hierarchy of logical connectives. • Construct truth tables by using an alternative method.

  3. Truth Tables A truth table is a diagram in table form that is used to show when a compound statement is true or false based on the truth values of the simple statements that make up the compound statement.

  4. Negation According to our definition of a statement, a statement is either true or false, but never both. Consider the simple statement p = “Today is Tuesday.” If it is in fact Tuesday, then p is true, and its negation (~p) “Today is not Tuesday” is false. If it’s not Tuesday, then p is false and ~p is true. The truth table for the negation of p looks like this. First write the possible conditions for p. It can be True or False. The negation ~p has the opposite truth values.

  5. Truth Tables with Two Simple Statements If we have a compound statement with two component statements p and q, there are four possible combinations of truth values for these two statements: Possibilities Symbolic value of each 1. p and q are both true. 2. p is true and q is false. 3. p is false and q is true. 4. p and q are both false.

  6. Conjunction (And) Suppose a friend tells you, “I bought a new computer and a new iPod.” This compound statement can be symbolically represented by p ∧ q, where p = “I bought a new computer.” q = “I bought a new iPod.” When would this conjunctive statement be true? • If your friend actually had made both purchases, then of course the statement would be true. • On the other hand, suppose your friend bought only a new computer or only a new iPod, or maybe neither of those things. Then the statement would be false.

  7. Truth Values for a Conjunction The conjunction p ∧ q is true only when both p and q are true. The truth table below summarizes the possibilities for the conjunction, “I bought a new computer and a new iPod.” p = “I bought a new computer.” q = “I bought a new iPod.” Bought computer and iPod Bought computer, not iPod Bought iPod, not computer Bought neither

  8. Disjunction (Or) Suppose your friend actually said, “I bought a new computer or a new iPod.” This compound statement can be symbolically represented by p ∨ q, where p = “I bought a new computer.” q = “I bought a new iPod.” When would this disjunctive statement be true? • If your friend actually did buy one or the other, or both, then this statement would be true. • And if he or she bought neither, then the statement would be false.

  9. Truth Values for a Disjunction The disjunction p ∨ q is true when either p or q or both are true. It is false only when both p and q are false. The truth table below summarizes the possibilities for the conjunction, “I bought a new computer or a new iPod.” p = “I bought a new computer.” q = “I bought a new iPod.” Bought computer and iPod Bought computer, not iPod Bought iPod, not computer Bought neither

  10. Conditional (If…then) A conditional statement, which is sometimes called an implication, consists of two simple statements using the connective if . . . then. The first component is called the antecedent. The second component is called the consequent. Think about the following simple example: “If it is raining, then I will take an umbrella,” symbolically written p → q, where p = “It is raining.” q = “I will take an umbrella.”

  11. Conditional (If…then) p → q p = “It is raining.” q = “I will take an umbrella.” We’ll break this down into four cases: • Case 1: It is raining and I do take an umbrella. Since I am doing what I said I would do in case of rain, the conditional statement is true. • Case 2: It is raining and I do not take an umbrella. Since I am not doing what I said I would do in case of rain, I’m a liar and the conditional statement is false. • Case 3: It is not raining and I do take an umbrella. I never said in the original statement what I would do if it were not raining, so we consider the original statement to be true. • Case 4: It is not raining, and I do not take my umbrella. This is essentially the same as case 3—I never said what I would do if it did not rain, so we consider the original statement to be true.

  12. Truth Values for a Conditional The conditional statement p → q is false only when the antecedent p is true and the consequent q is false. The truth table below summarizes the possibilities for the conditional, “If it is raining, then I will take an umbrella.” p = “It is raining.” q = “I will take an umbrella.” Raining, take umbrella Raining, do not take umbrella Not Raining, take umbrella Not Raining, do not take umbrella

  13. Biconditional (If and only if) A biconditional statement is really two statements in a way; it’s the conjunction of two conditional statements. In symbols, we can write either p ↔ q or (p → q) ∧ (q → p). Since the biconditional is a conjunction, for it to be true, both of the statements p → q and q → p must be true.

  14. Biconditional (If and only if) We’ll also break p ↔ q down into four cases: • Case 1: Both p and q are true. Then both p → q and q → p are true, and the conjunction (p → q) ∧(q → p), which is also p ↔ q, is true as well. • Case 2: p is true and q is false. In this case, the implication p → q is false, so it doesn’t even matter whether q → p is true or false—the conjunction has to be false. • Case 3: p is false and q is true. This is case 2 in reverse. The implication q → p is false, so the conjunction must be as well. • Case 4: p is false and q is false. According to the truth table for a conditional statement, both p → q and q → p are true in this case, so the conjunction is as well.

  15. Truth Values for a Biconditional The biconditional statement p ↔ q is true when p and q have the same truth value and is false when they have opposite truth values.

  16. EXAMPLE 1 Constructing a Truth Table Construct a truth table for the statement ~p ∨ q. SOLUTION Step 1 Set up a table as shown. The order in which you list the Ts and Fs doesn’t matter as long as you cover all the possible combinations. For consistency, we’ll always use the pattern shown. Step 2 Find the truth values for ~p by negating the values for p, and put them into a new column, column 3, marked ~p. Truth values for ~p are opposite those for p.

  17. EXAMPLE 1 Constructing a Truth Table Construct a truth table for the statement ~p ∨ q. SOLUTION Step 3 Find the truth values for the disjunction ~p ∨ q. Use the T and F values for p and q in columns 2 and 3, remembering that an disjunction is false only when both components are false. The truth values for the statement ~p ∨ q are found in column 4. The statement is true unless p is true and q is false. “Or” is false only when both are false. The statement is only false when p = T and q = F.

  18. EXAMPLE 2 Constructing a Truth Table Construct a truth table for the statement ~(p → ~q). SOLUTION Step 1 Set up a table as shown. Step 2 Find the truth values for ~q by negating the values for q, and put them into a new column marked ~q. Step 3 Find the truth values for the implication p → ~q, using the values in columns 1 and 3, remember that an implication is false only when the antecedent is true and the consequent is false. Truth values for ~q are opposite those for q. “If…then” is false only when “if” is true and “then” is false.

  19. EXAMPLE 2 Constructing a Truth Table Construct a truth table for the statement ~(p → ~q). SOLUTION Step 4 Find the truth values for the negation ~(p → ~q) by negating the values for p → ~q in column 4. The truth values for ~(p → ~q) are in column 5. Truth values for ~(p → ~q) are opposite those for p → ~q. The statement is only true when p and q are true.

  20. EXAMPLE 3 Constructing a Truth Table Construct a truth table for the statement p ∨ (q → r). SOLUTION Step 1 Set up a table as shown. The order in which you list the Ts and Fs doesn’t matter as long as you cover all the possible combinations. For consistency, we’ll always use the pattern shown for 3 letters. Step 2 Find the truth value for the statement in parentheses, q → r.

  21. EXAMPLE 3 Constructing a Truth Table Construct a truth table for the statement p ∨ (q → r). SOLUTION Step 4 Find the truth values for the disjunction p ∨(q → r), using the values for p from column 1 and those for q → r from column 4. The truth values for the statement p ∨ (q → r) are found in column 5. The statement is true unless p and r are false while q is true.

  22. Hierarchy of Connectives We have seen that when we construct truth tables, we find truth values for statements inside parentheses first. To avoid having to always use parentheses, a hierarchy of connectives has been agreed upon by those who study logic. 1. Biconditional ↔ 2. Conditional → 3. Conjunction ∧, disjunction ∨ 4. Negation ~ When we find the truth value for a compound statement without parentheses, we find the truth value of a lower-order connective first. For example, p ∨q → r is a conditional statement since the conditional (→) is of a higher order than the disjunction (∨). If you were constructing a truth table for the statement, you would find the truth value for ∨ first.

  23. EXAMPLE 4 Using the Hierarchy of Connectives For each, identify the type of statement using the hierarchy of connectives, and rewrite by using parentheses to indicate order. (a) ~p ∨ ~q (b) p → ~q ∧ r (c) p ∨q ↔ q ∨r (d) p → q ↔ r

  24. EXAMPLE 4 Writing Statements Symbolically SOLUTION (a) For ~p ∨ ~q : the ∨ is higher than the ~; the statement is a disjunction and looks like (~p) ∨ (~q) with parentheses. (b) For p → ~q ∧ r : the → is higher than the ∧ or ~; the statement is a conditional and looks like p → (~q ∧r) with parentheses. (c) For p ∨q ↔ q ∨r : the ↔ is higher than ∨; the statement is a biconditional and looks like (p ∨q) ↔ (q ∨r) with parentheses. (d) For p → q ↔ r : the ↔ is higher than the →; the statement is a biconditional and looks like (p → q) ↔ r with parentheses.

  25. EXAMPLE 5 An Application of Truth Tables Use the truth value of each simple statement to determine the truth value of the compound statement (p ∨q) → r, if… p: O. J. Simpson was convicted in California in 1995. q: O. J. Simpson was convicted in Nevada in 2008. r : O. J. Simpson gets sent to prison.

  26. EXAMPLE 5 Translating Statements from Symbols to Words SOLUTION In probably the most publicized trial of recent times, Simpson was acquitted of murder in California in 1995, so statement p is false. In 2008, however, Simpson was convicted of robbery and kidnapping in Nevada, so statement q is true. Statement r is also true, as Simpson was sentenced in December 2008. Since we know the truth values of each simple statement, then we only need to analyze the case when p = F, q = T and r = T. (p ∨q) → r substituting the truth values (F ∨ T) → T The disjunction is true, so T → T Leading to the implication being true. T

  27. EXAMPLE 6 Constructing a Truth Table Using an Alternative Method Construct a truth table for the statement ~(p → ~q). SOLUTION Step 1 Set up a table as shown. Step 2 Write the truth values for p and q under their respective letters in the statement as shown, and label the columns as 1 and 2. Step 3 Find the negation of q since it is inside parentheses, and place the truth values in column 3, just below the negation symbol. 3 2 1 Draw a line through the truth values in column 2 since they will not be used again.

  28. EXAMPLE 6 Constructing a Truth Table Using an Alternative Method Construct a truth table for the statement ~(p → ~q). SOLUTION Step 4 Find the truth values for the conditions (→)by using the T and F values in columns as 1 and 3. Place the values in column 4 and draw a line through columns 1 and 3. Step 3 Find the negations of the truth values in column 4, since we’re now focused on the negation outside of parentheses, and place the values in column 5. 5 4 3 2 1 Note that the values in column 5 are the same as in Example 2.

  29. EXAMPLE 7 Constructing a Truth Table Using an Alternative Method Construct a truth table for the statement p ∨ (q → r). SOLUTION Step 1 Set up a table as shown. Step 2 Recopy the values of p, q, and r under their respective letters in the statement as shown & number the columns. Step 3 Find the conditional using the truth values in 2 & 3. Place them under the → and label it column 4. 2 4 3 1

  30. EXAMPLE 7 Constructing a Truth Table Using an Alternative Method Construct a truth table for the statement p ∨ (q → r). SOLUTION Step 4 Complete the truth table for the disjunction, using the truth values in columns 1 and 4. The truth values for p ∨ (q → r) are found in column 5. These are the same values we found in Example 3. 5 2 4 3 1

More Related